Rate Equations for Graphs

Abstract

In this paper, we combine ideas from two different scientific traditions: 1) graph transformation systems (GTSs) stemming from the theory of formal languages and concurrency, and 2) mean field approximations (MFAs), a collection of approximation techniques ubiquitous in the study of complex dynamics. Using existing tools from algebraic graph rewriting, as well as new ones, we build a framework which generates rate equations for stochastic GTSs and from which one can derive MFAs of any order (no longer limited to the humanly computable). The procedure for deriving rate equations and their approximations can be automated. An implementation and example models are available online at https://rhz.github.io/fragger. We apply our techniques and tools to derive an expression for the mean velocity of a two-legged walker protein on DNA.

Transmission dynamics of SARS-CoV-2 in a strictly-Orthodox Jewish community in the UK

“…We used an extension of the κ-calculus (29), in κ language for this study, which is a rule-based stochastic process calculus that is best understood as a graph rewriting system (30,31) for labelled site graphs. It has been extensively used in its original form in molecular biology to study the dynamics of interacting large molecules or polymers, and has been shown to be applicable to population biology and epidemics (45).…”

“…We constructed a transmission model using an extended version of the kappa-calculus (29) to implement a transmission model as a stochastic graph rewriting system (30,31), a generalalisation of how explicit epidemic dynamics are usually formulated on networks (32). In this model, individuals have disease progression states and transmission is mediated by place, with a separate transmission process for each setting or kind of place.…”

Rewriting Theory for the Life Sciences: A Unifying Theory of CTMC Semantics

“…In particular, albeit a rewriting-based graphical formalism, Kappa as originally introduced in [23] is not a categorical rewriting formalism, although categorical approaches have been employed to model certain aspects of its semantics [27]. This renders comparing this approach with the computation of commutators presented in this paper highly intricate.On the other hand, an interesting line of work by Danos et al [28,29] demonstrated that at least in the less technically involved setting of rewriting over adhesive categories without conditions or constraints, some of the syntactic methods utilized in the Kappa framework could be reinterpreted in the adhesive rewriting setting in order to obtain algorithms for computing first-order moment ODEs for pattern counting observables 8 , providing a first hint at the possible existence of a general rewriting-based CTMC formalism for Kappa.…”